Just for fun, here's another approach, based on the idea of XOR-ing
the string with shifted versions of itself, to then extract the
recurring substring from the case with the longest trailing "nulled-out"
part. Hope you'll appreciate the creative aspect :) — even though
it's not as compact as the regex approach suggested by ikegami. Due to its generic approach, the algorithm should work not
only with numbers, but with any kind of string.

BTW, I'll admit it up-front: in its current version, this algorithm
has a subtle problem. As you can see when looking at the last number
(1/467), it falsely reports '7' as the recurring part, as a result of
the number being cut off prematurely due to insufficient precision...
(a problem it shares with the regex solution, btw). Presumably some
heuristic workaround can be found for that, but I'll leave this as an
'exercise for the reader'... ;)